# MORITZ L. FRANKENHEIM (1801 - 1869).*Crystallonomische Aufsätze*. (*Isis von Oken* **1826**, columns 497-515 and 542-565.)

Frankenheim was born in 1801 in Brunswick, and studied physics in college. After obtaining his degree in 1823 with a dissertation on the theory of gases and vapors, he shifted his research interests to crystallography. In 1827 he obtained an appointment at the University of Breslau, where he was assistant professor of physics, geography, and mathematics from 1827 to 1850, and then full professor of these subjects starting in 1850. After his retirement in 1866, he lived in Leipzig and then Dresden.

Frankenheim made significant contributions to the theoretical understanding of crystal shapes and to the mathematics of the symmetry of crystals. In his 1826 paper* Crystallonomische Aufsätze (Crystal Geometry Essays)* he classified all the possible symmetries for the shapes of crystals, and in doing this was the first to state correctly that there were exactly 32 such symmetries, now called crystallographic point groups. In the same 1826 paper, Frankenheim independently proposed an improved way to describe the spatial orientations of crystal faces by means of a sequence of three whole numbers; this same method had been published the year before by the Englishman William Whewell (1794 - 1866). Some years later, the British crystallographer William Miller popularized this method of describing the relative orientations of crystal faces, and somewhat unfairly the three whole numbers became known as Miller indices.

It has been known since antiquity that arrays of points in space (called lattices) can have only two-, three-, four-, or six-fold rotational symmetry (this fact is why bathroom floors cannot be tiled with regular pentagons). From this fact, however, Frankenheim deduced something new: that there are no more than 15 different types of three-dimensional lattices. Twenty five years later, the French crystallographer Auguste Bravais corrected the number to 14 (Frankenheim had mistakenly counted one type twice). In addition to this theoretical work, Frankenheim conducted some of the first microscopic examinations of crystals in polarized light.

Unfortunately, Frankenheim's crystallographic achievements (and the credit that should have been his) were unknown to the scientific community until his work was "rediscovered" in the 20th century.

Curtis P. Schuh, *Mineralogy & Crystallography: An Annotated Bibliography of Books Published 1469 through 1919.* Tucson: privately published, 2005, p540-541.